The Milnor–Moore Theorem
This post is about the Milnor–Moore theorem, a powerful tool describing the structure of (co)commutative Hopf algebras. Like the Eckmann–Hilton argument, it shows that having multiple compatible operations on the same object can lead to unexpected results about the object. Briefly, the theorem says that as soon as the Hopf algebra is cocommutative and connected, then it is isomorphic to the universal enveloping algebra of a Lie algebra (and a similar dual statement is true for commutative Hopf algebras).
As the name indicates, the theorem is due to Milnor and Moore in the paper cited below. The details of this post will mostly be based on the Chapter 7 of the book of Fresse cited below, and if there’s no reference for a theorem or a proposition, you can find it there. As usual, I mostly wanted to write this post because I often find myself forgetting how the proof of the theorem goes, and hopefully writing for a general audience it will fix it in my mind.
Algebras and coalgebras
A Hopf algebra is the combination of two structures: associative algebra and coassociative coalgebra. Let’s recall what that means. From now on we assume that we are working over some field ; later, we will assume that this field has characteristic zero.
Definition. A (unital) associative algebra is a vector space equipped with a product and a unit satisfying:
If we write and , then these two axioms merely say that and .
The definition of a coalgebra is more-or-less formally dual:
Definition. A coassociative coalgebra is a vector space equipped with a coproduct and a counit satisfying:
We will use Sweedler’s notation: for , we write
The counitality axiom then becomes, for example, .
We will immediately switch to the differential graded (dg) setting. A graded vector space is a vector space equipped with a decomposition . A differential on such a vector space is a linear map of degree (i.e. ) that satisfies .
A graded algebra is a graded vector space equipped with the structure of an algebra compatible with the grading: . A derivation on a graded associative algebra is a map of degree such that (this is the last time I’ll write an explicit sign, from now on I’ll use the Koszul rule of signs). A dg-algebra is, finally, a graded algebra equipped with a derivation satisfying . The definition of a dg-coalgebra is similar (just add “co-” in front of every word).
Example. The base field itself has both the structure of an algebra and a coalgebra, given by the canonical isomorphism , and the identity for the (co)unit.
Bialgebras and Hopf algebras
Definition. A (dg-)bialgebra is a dg-vector space equipped with the structure of an algebra and the structure of a coalgebra such that the coproduct and the counit are morphisms of algebras, where is given its canonical algebra structure (or equivalently, the product and the unit are morphisms of coalgbras).
A Hopf algebra is a bialgebra equipped with a linear endomorphism satisfying, for all :
This is a lot of structure! There’s a product, a unit, a coproduct, a counit, and an antipode, satisfying a whole bunch of relations. If it exists, the antipode is unique, but its existence is not guaranteed. Fortunately, most of the time the antipode comes for free:
Theorem. Let be a bialgebra, and suppose that is connected, i.e. for and (and the (co)unit are the identity). Then there exists an antipode making into a Hopf algebra.
The tensor algebra on some dg-module is given by:
with grading and differential induced by the grading and the differential of ( is put in degree 0 and has trivial differential). The product is given by concatenation of tensors:
and the unit is . Then is the free associative algebra on : for all algebras and dg-linear morphism , there exists a unique dg-algebra morphism lifting (through the obvious inclusion ).
One can then define a Hopf algebra structure on : the counit lifts , the coproduct lifts , , and the antipode lifts , . It’s possible to explicitly describe the coproduct using shuffles:
Note that the coproduct is cocommutative, but the product is not commutative.
The tensor coalgebra on some dg-module is also given by:
The underlying dg-module is the same, but the Hopf algebra structure is different. Now it’s the coproduct that’s described more easily: it is given by deconcatenation of tensors,
The counit is again given by if . Then is the cofree conilpotent coassociative coalgebra on : for every conilpotent coalgebra and every dg-linear morphism , there exists a unique dg-coalgebra morphism lifting through the obvious projection . (A fun exercise.)
The product and the unit are defined similarly as for , and the product is again described using shuffles; it is commutative.
The symmetric algebra is the quotient of the tensor algebra by the ideal generated by tensors of the forms . It is clearly graded commutative, and the coproduct factors through the quotient, giving a Hopf algebra structure that is at the same time commutative and cocommutative.
The symmetric coalgebra is, on the other hand, given by invariants: is the module of tensors invariant by the action of the symmetric groups. The product and coproduct factor through the inclusion, and moreover the coproduct becomes cocommutative when restricted to : it is also a commutative and cocommutative Hopf algebra. In characteristic zero, and are in fact isomorphic using the trace map.
Structure of Hopf algebras
Primitive elements and indecomposable
From now on, we let be some Hopf algebra.
Definition. An element is said to be primitive if and . The set of primitive elements is .
Proposition. The set of primitive elements is a Lie algebra, with bracket given by the commutator .
This is not very hard to check. The functor of primitive elements is in fact right adjoint to the functor of universal enveloping algebras.
Proposition. The inclusion induces isomorphism , where is endowed with the abelian Lie algebra structure. The inclusion induces an isomorphism between , the free Lie algebra on , and .
This gives a concrete way of defining the free Lie algebra.
We can do a dual construction with indecomposables. The augmentation ideal of is (more generally, this is defined for an augmented algebra). The product of defines a map on the quotient , and we can define:
Definition. The module of indecomposables is the quotient .
Proposition. The dg-module is a Lie coalgebra, with cobracket given by the antisymmetrisation of the coproduct of .
The verification of this is formally dual to the proof of the proposition about primitive elements, and is left adjoint to the functor of universal coenveloping coalgebras.
Proposition. The projection induces an isomorphism , where is endowed by the abelian Lie coalgebra structure. The projection induces an isomorphism from to , the cofree Lie coalgebra on .
The theorem of Milnor–Moore
Let us now assume that the base field has characteristic zero. We will not state the Milnor–Moore theorem in full generality: I will assume the restrictive hypothesis that is connected and has finite type, but the theorem applies more generally to locally conilpotent Hopf algebras.
Theorem [Milnor–Moore]. Let be a connected, cocommutative Hopf algebra of finite type. Then the inclusion induces an isomorphism of Hopf algebras .
One also has the dual theorem:
Theorem [Milnor–Moore]. Let be a connected, commutative Hopf algebra of finite type. Then the quotient map induces an isomorphism of Hopf algebras .
Now the proof (of which I will just give a sketch) is rather nice. I’ll more-or-less follow the original proof of Milnor–Moore. It works by induction, which is easier to understand in the dual case. We will first prove that , then conclude by the Poincaré–Birkhoff–Witt theorem.
The first isomorphism is clear if the Hopf algebra only has a single generator (i.e. is one-dimensional). Now if , then one can quotient out by the sub-algebra generated by the first indecomposables to get . The quotient has a single generator, and the sub-algebra has generators, so it is enough to show that is isomorphic as an algebra to the tensor product of the subalgebra and the quotient.
The quotient map has a linear section (which isn’t necessarily a morphism of Hopf algebras). This yields a map . And now the heart of the proof is in proving that this map is an isomorphism of algebras using the Hopf algebra structure. It is used to choose the section wisely enough so that the resulting map is an isomorphism of algebras.
Now the (dual) Poincaré–Birkhoff–Witt theorem says that is an isomorphism of algebras. The isomorphism (which is explicit) fits in a commutative triangle with the isomorphism just constructed and the canonical morphism of Hopf algebras . Using the 2-out-of-3 property of isomorphisms, this last map is thus an isomorphism (of Hopf algebras) .